Binary operations


Set is a collection of definite well defined objects.

Set is denoted by capital letter.

For example:

= {a, b, c, d, e}


à a non-empty set

GXG = {(a ,b): a  G, b  G }.

Above line is read as: G cross G equal to (a, b) such that ‘a’ belongs to ‘G’, ‘b’ belongs to ‘G’.


f : GXG = G,

Above line is read as ‘f’ is such that G cross G is equal to G.

Here ‘f’ is an operation of ‘X’ on two groups ‘G’ and ‘G’.

The output of ‘GXG’ is also a ‘ G ‘ so this type of operation is known as Binary Operation on a set G.


Operation ‘f’ on ‘G’ and ‘G’ can be denoted as ‘GfG’ , or ‘afb’ where (a  G, b   G).

+, x, etc symbols are used in Binary Operations.

Binary Operations examples:-

      1)      a + b   G,  a, b  G.

Above line is read as ‘a’ plus ‘b’ belongs to ‘G’, for all ‘a’, ‘b’ belongs to ‘G’.

Here, ∀ à  for all.

      2)      a * b = G,  a,b  G.

      3)      Addition of natural numbers is also a natural number.

Natural number are also known as all non-negative or positive numbers (0,1,2,3,4……).

If, N à Set of natural numbers

A + b  N,  a, b  N.

Above line is read as ‘a’ plus ‘b’ belongs to ‘N’, for all ‘a’, ‘b’ ‘N’.

      4)      Subtraction is not binary operation on N (natural numbers).

Nà Set of natural numbers.

3 – 5 = - 2  N, whereas 3, 5  N.

Above line is read as three minus five is not belongs to ‘N’, whereas three, five belongs to ‘N’.

      5)      Subtraction is binary operation on I (integer numbers).

I --> Set of integer numbers

Python Programming ↓ 👆
Java Programming ↓ 👆
JAVA covered following topics in these notes.
JAVA Programs
Principles of Programming Languages ↓ 👆
Principles of Programming Languages covered following topics in these notes.

Previous years solved papers:
A list of Video lectures References:
  1. Sebesta,”Concept of programming Language”, Pearson Edu 
  2. Louden, “Programming Languages: Principles & Practices” , Cengage Learning 
  3. Tucker, “Programming Languages: Principles and paradigms “, Tata McGraw –Hill. 
  4. E Horowitz, "Programming Languages", 2nd Edition, Addison Wesley

    Computer Organization and Architecture ↓ 👆

    Computer Organization and Architecture covered following topics in these notes.

    1. Structure of desktop computers
    2. Logic gates
    3. Register organization
    4. Bus structure
    5. Addressing modes
    6. Register transfer language
    7. Direct mapping numericals
    8. Register in Assembly Language Programming
    9. Arrays in Assembly Language Programming


    1. William stalling ,“Computer Architecture and Organization” PHI
    2. Morris Mano , “Computer System Organization ”PHI

    Computer Network ↓ 👆
    Computer Network covered following topics in these notes.
    1. Data Link Layer
    2. Framing
    3. Byte count framing method
    4. Flag bytes with byte stuffing framing method
    5. Flag bits with bit stuffing framing method
    6. Physical layer coding violations framing method
    7. Error control in data link layer
    8. Stop and Wait scheme
    9. Sliding Window Protocol
    10. One bit sliding window protocol
    11. A protocol Using Go-Back-N
    12. Selective repeat protocol
    13. Application layer
    1. Andrew S. Tanenbaum, David J. Wetherall, “Computer Networks” Pearson Education.
    2. Douglas E Comer, “Internetworking with TCP/IP Principles, Protocols, And Architecture",Pearson Education
    3. KavehPahlavan, Prashant Krishnamurthy, “Networking Fundamentals”, Wiley Publication.
    4. Ying-Dar Lin, Ren-Hung Hwang, Fred Baker, “Computer Networks: An Open Source Approach”, McGraw Hill.